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Topic: Re: Calculating limits, again
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Posts: 10,730
Registered: 12/3/04
Re: Calculating limits, again
Posted: Aug 11, 1997 5:12 PM
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In a message dated 97-08-11 16:24:26 EDT, Jerry Rosen writes, quoting Mark
Snyder and me,

> >I really like all these suggestions for doing the limit of radical x -
> >x, but I guess the real question is WHY are you doing the limit?
> Well, *I* did it because I was asked to...

> >Possible answers:
> >
> >(1) You need to know the answer to do some other problem. In that case

> >it on your calculator and get on with the other problem.
> >(2)You are a good teacher and want to show your class how to do limits (a)
> >using algebraic manipulation, (b) using a Taylor series, (c) using a table

> >values (d) using Mathematica or (e) etc. In this case use (a) algebraic
> >manipulation, (b) a Taylor series, (c) a table of values (d) Mathematica

> >(e) etc.


> (1) doesn't actually work. Graphing something on your calculator only
> gives you an approximate answer, or a hint as to what the real answer is.
> The same thing holds for numerical investigations of limits. Just because
> the graph "looks like" it hits the number 1, or the number "seems to be
> settling down to 1" doesn't mean that it does. It might hit the number
> 1.00001, or 1 - (1/pi^89), and depending on the context that gave rise to
> the problem, that difference might be significant. Or maybe it settles
> down to 1 until you get to within 10^(-50) of the value, and then it jumps
> up to pi^2/6. >>

I know all that. I know you can find examples where a calculator fails. BUT
then you go on to talk of engineers and exact answers in the same paragraph.
Is it important to an engineer in a practical situation if the limit is 1 or
1 - (1/pi^89) ??? There are of course calculator and CAS on computers (same
thing) that find the "exact" value.

Graphing (or numerical analysis) is of little, if any, help when you have a
> function that depends on two (or more) variables, and you want the limit as
> one of the variables approaches, say, infinity. This is *extremely* common
> in physics problems. In that case, you are interested in what the limiting
> *function* is, and algebraic tricks, LHop, and Taylor series are the only
> approaches that give you that.


Of course if you can't find this particular limit on a calculator (or CAS)
then you can't find it on a calculator (or CAS) and you'll have to find it
some other way. BTW the TI92 can find limits as x approaches infinity, as
can may computer programs.


And I think that there is a (4)

> (4) You are a good teacher who has misplaced the phone numbers of the
> people who your students will be working with ten years from now, so you
> can't call them up to ask which approach each of your students will need,
> so you cover all the approaches (even to the point of having your students
> think you are just "showing off"). This way they will be prepared for
> that, as well as for the AB test.


Your number 4 is my number 2. Certainly a good teacher will show all the
methods mentioned. That was the point of my original question " WHY are you
doing the limit?"

Lin McMullin
Ballston Spa, NY

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